For finite lists, Cartesian lifting of a binary operation to lists preserves associativity. Both associations enumerate triples in the same order: the entry from xs changes slowest, then the entry from ys, then the entry from zs changes fastest. The results are respectively
(1)
and
(2)
Associativity makes corresponding entries equal; their order and multiplicity already agree. Empty lists make both sides empty. Therefore
(3)

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